New neutrino physics and the altered shapes of solar neutrino spectra
Ilídio Lopes∗
arXiv:1702.00447v1 [astro-ph.SR] 1 Feb 2017
Centro Multidisciplinar de Astrofísica, Instituto Superior Técnico,
Universidade de Lisboa , Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Neutrinos coming from the Sun’s core are now measured with a high precision, and fundamental neutrino
oscillations parameters are determined with a good accuracy. In this work, we estimate the impact that a new
neutrino physics model, the so-called generalized Mikheyev-Smirnov-Wolfenstein (MSW) oscillation mechanism, has on the shape of some of leading solar neutrino spectra, some of which will be partially tested by
the next generation of solar neutrino experiments. In these calculations, we use a high-precision standard solar
model in good agreement with helioseismology data. We found that the neutrino spectra of the different solar
nuclear reactions of the proton-proton chains and carbon-nitrogen-oxygen cycle have quite distinct sensitivities
to the new neutrino physics. The HeP and 8 B neutrino spectra are the ones for which their shapes are more
affected when neutrinos interact with quarks in addition to electrons. The shape of the 15 O and 17 F neutrino
spectra are also modified, although in these cases the impact is much smaller. Finally, the impact in the shape
of the P P and 13 N neutrino spectra is practically negligible.
I.
INTRODUCTION
Since their discovery in 1956, neutrinos have always surprised physicists due to their unexpected properties, often
challenging our basic understanding of the standard model
of particle physics (in the remainder of the article, it will
be called simply the ’standard model’) and the properties
of elementary particles. In particular, the discovery of neutrino flavour oscillations stands as one of the most convincing
proofs that the standard model is incomplete as it does not
explain all the known experimental properties of the fundamental particles.
The neutrino research success has been made possible
mostly due to many dedicated experiments performed during the last fifty years. It is worth highlighting the contributions of some pioneering experiments, among others, such
as the Super-Kamiokande detector [1, 2] where the oscillation of atmospheric neutrinos was discovered, and the SNO
detector [3] where the fluxes of all neutrino flavour species
produced in the Sun’s core were measured for the first time.
Many other experiments done during the previous decades
have contributed for the success of this story, in particular,
the solar neutrino experiments. Despite their technical complexities, these experiments were able to measure the electron
neutrino fluxes coming from the Sun and played a major role
in the establishment of the so-called solar neutrino problem –
a discrepancy between the theoretical prediction of neutrino
fluxes and their experimental measurements: the experimental value being one third of the predicted value. This fact was
evidenced for the first time by the Homestake Experiment of
Ray Davis [4], and confirmed by many other experiments that
followed. It was the solar neutrino problem that prompted the
development of the neutrino flavour oscillation model.
If indeed the previous generation of solar neutrino detectors has been one of the beacons of particle physics, both by
leading the way in uncovering the basic properties of particles, including the nature of neutrino flavour oscillations, and
∗
[email protected]
by being responsible for developing pioneering techniques in
experimental neutrino detection [5], the next generation of
detectors is equally promising in discovering new physics.
Among various, some of which will be looking for evidence
of neutrino new physics we can mention the following future
detectors: the Low Energy Neutrino Astronomy [LENA, 6],
the Jiangmen Underground Neutrino Observatory [JUNO, 7],
the Deep Underground Neutrino Experiment [DUNE, 8], the
NOνA Neutrino Experiment [NOνA, 9], and the Jinping Neutrino Experiment [Jinping, 10].
These detectors will measure with high precision the neutrino fluxes and neutrino spectra of a few key neutrino nuclear
reactions, such as the 8 B electron-neutrino (8 Bνe ) spectrum
produced by the β-decay process in the 8 B solar (chain) reaction: 7 Be(p, γ)8 (e+ νe )8 B ∗ (α)4 He [11, 12]. This will allow us to probe in detail the Sun’s core, including the search
for new neutrino physics interaction or even new physics processes. Moreover, the high quality of the data will enable
the development of inversion techniques for determining basic properties of the solar plasma [e.g., 13]. Specific examples
can be found in Balantekin et al. [14] and Lopes [15]. Equally,
solar neutrino data can be used to find specific features associated with possible new physical processes present in the Sun’s
interior [e.g. 16], such as the possibility of an isothermal solar
core associated with the presence of dark matter [17].
Today, the basic principles of neutrino physics are firmly
established, neutrinos are massive particles with a lepton
flavours mix. The parameters describing neutrino flavour oscillations are measured with great accuracy and precision,
which has been possible due to the extensive studies made
by many different types of neutrino experiments: solar and
atmospheric neutrino observatories, nuclear reactors and experimental particle accelerators [e.g., 18–20]. Section III C
presents the status of the current neutrino oscillation parameters obtained from up-to-date experimental data.
Even if many properties of neutrinos are known, many others are still a mystery:
- firstly, are neutrinos Majorana or Dirac fermions ? i.e.,
are neutrinos their own anti-particle ? Although the theoretical expectation favours the first option, only experimental ev-
2
idence can settle this question;
- secondly, what is the mass hierarchy of neutrinos ? In
other words, does the order of neutrino masses between the
different particle families follow a normal hierarchy – two
light neutrinos followed by a heavier one, or an inverted hierarchy – one light neutrino follow by two heavier ones ?
18
Together with the CP violation in the lepton sector, these
are the most important questions of neutrino physics. Some
of these questions will be answered by the next generation of
neutrino experiments – the long baseline neutrino experiments
and solar neutrino telescopes. Nevertheless, it is necessary to
improve the current neutrino flavour model to take full advantage of the forthcoming experimental data.
10
Despite the success of the current neutrino physics model in
explaining most of the neutrino’s known observed properties,
the solution encountered clearly indicate the existence of new
physics beyond the standard model. As such, this implies that
within the current particle physics theoretical framework, experiments can study neutrinos in other types of interactions.
When such processes occur, these lead to important modifications of the physical mechanisms by which neutrinos are
created, propagate and interact with other particles of the standard model. This new class of neutrino interactions is usually
known as non-standard interactions (nsi).
The non-standard interactions of neutrinos have been extensively studied in the literature, among others reviews on this
topic, see for instance Miranda and Nunokawa [21], Ohlsson
[22]. Moreover, the constraints on the nsi parameters and
their effects for low energy neutrinos have been derived from
a great variety of experimental results. Until now no definitive evidence of non-standard interactions has been provided
by the experimental data. Actually, all observations made as
yet can be explained in terms of the standard interactions of
the three known neutrinos, although some of them need the
help of sterile neutrinos. Nevertheless, in some cases the nonstandard interactions of neutrinos provide an interesting and
valid alternative [e.g., 23]
In this work we are mostly concerned with the non-standard
interactions of solar neutrinos. These interactions can affect
the neutrino production inside the Sun, the detection of neutrinos by experimental detectors and the neutrino propagation
in the Earth’s and Sun’s interiors. In particular, our study focus on the propagation of neutrinos through baryonic matter in
the Sun’s interior, a process usually known as the generalized
Mikheyev-Smirnov-Wolfenstein (M SW ) oscillation mechanism, or generalized matter effect oscillations. Our goal is to
make predictions about the modifications imprinted by this
new generalized M SW on the shape of the solar neutrino
spectrum produced by some of the (pp) and carbon-nitrogenoxygen (CNO) key nuclear reactions, like HeP and 8 B neutrino spectra.
The high quality of the standard solar model in reproducing
the measured solar neutrino fluxes, and the observed acoustic frequency oscillations, make it a privileged tool to look for
the new interactions within a generalized Mikheyev-SmirnovWolfenstein mechanism occurring in the Sun’s interior. The
standard solar model [SSM, 24] partly validated by helioseis-
16
14
PP
PeP(*)
HeP
7Be(*)
8B
13N
15O
17F
Φ (r)
12
8
6
4
2
0
0
0.05
0.1
0.15
0.2
0.25
Radius (solar radius)
0.3
0.35
FIG. 1. The electron-neutrino fluxes produced in the various nuclear reactions of the pp chains and CNO cycle.These neutrino fluxes
were calculated for a standard solar model using the most updated
microscopic physics data. This solar model is in agreement with
the most current helioseismology diagnostic and other solar standard models published in the literature (see text). For each neutrino
type j (with j = P P, P eP (∗), HeP,8 B,7 Be(∗),13 N,15 O,17 F ),
Φj (r) ≡ (1/Fj ) dfj (r)/dr is drawn as a function of the fractional
radius r for which fj is the flux in s−1 and Fj is the total flux for this
neutrino type. The neutrino sources noted with the symbol (∗) correspond to spectral lines. The same colour scheme is used in figures 4
and 5.
mology, predicts that the density inside the Sun varies from
about 150 g cm−3 in the centre of the star, to 1 g cm−3 at half
of the solar radius. The variation of density of matter with
the solar radius is followed by identical variations on the local quantities of electrons and quarks. Moreover, the different
type of quarks will also be affected by the local distribution of
chemical elements (most noticeably Hydrogen and Helium)
which lead to a not obvious distribution of up- and downquarks. Therefore, we can anticipate that the current standard
solar model combined with data coming from the next generation of the solar neutrino detections, will allow us to put
much stronger constraints in the non-standard interactions of
neutrinos.
In the next section, we review the current status of the standard solar model and neutrino production in the Sun’s core,
In Section III, we present a summarised discussion about the
current standard neutrino oscillation flavour model, and a generalized model for which neutrinos have new types of interactions with standard particles. In Section IV, we compute
the neutrino spectra resulting from these new types of interactions. In the final section we discuss the results and their
implications for the future neutrino experiments.
3
15
NEUTRINO PRODUCTION IN THE SUN’S CORE
A.
Helioseismology and the standard solar model
During the last three decades helioseismology has provided
solar physics with a tool that describes with unprecedented
quality the internal structure of the Sun from its surface up
to the deepest layers of the Sun’s interior. This has allowed
astronomers to characterise with great precision the different
solar neutrino sources. Equally, this discipline has stimulated
the development of inversion techniques to probe the internal solar dynamics. Today an impressive agreement has been
reached between the neutrino flux predictions and the neutrino
flux measurements made by the existing neutrino detectors.
The high quality of the helioseismology data has allowed to
compute an exceptionally accurate model of the Sun’s interior
- the standard solar model. The neutrino fluxes predictions of
the solar model have an accuracy comparable to the current
measurements made by particle accelerators or nuclear reactors.
The standard solar model in this study is obtained using
a version of the one-dimensional stellar evolution code CESAM [25]. The code has an up-to-date and very refined microscopic physics (updated equation of state, opacities, nuclear reactions rates, and an accurate treatment of the microscopic diffusion of heavy elements), including the solar
mixture of Asplund et al. [26, 27]. This solar model is calibrated to reproduce with high accuracy the present total radius, luminosity and mass of the Sun at the present t =
4.54 ± 0.04 Gyr [28]. Moreover, this model is required to
have a fixed value of the photospheric ratio (Z/X) , where X
and Z are the mass fraction of hydrogen and the mass fraction
of elements heavier than helium. This solar standard model
shows acoustic seismic diagnostics and solar neutrino fluxes
similar to other models found in the literature [24, 29–34]
This solar model is calibrated for the present day solar data
with a high accuracy. Therefore slightly different physical assumptions, will lead to different radial profiles of temperature,
density and chemical composition, among other quantities.
These changes result from readjustments of the Sun’s internal
structure caused by the need to obtain the same total luminosity. In particular, the neutrino fluxes and sound speed profile
will be very sensitive to the radial distributions of the previous quantities. As such, using the high precision data from
helioseismology, it is possible to put strong constraints to the
internal structure of the Sun and its neutrino fluxes [28, 35].
The current uncertainty between the square of the sound
speed profile inferred from helioseismology acoustic data and
the one obtained from the standard solar model using the upto-date photospheric abundances Asplund et al. [27] is smaller
than 3% for any layer of the Sun’s interior. Although there is
a difference between the sound speed profile computed using an older mixture of abundances by Grevesse and Sauval
[36] or the new mixture of Asplund et al. [27], for this study
these effects are negligible on the radial variation of electrons,
protons and neutrons. This is even more so, since recent measurements of the solar metallicity abundances suggest that the
ne(r) nu(r) nd(r) [1031 cm−3]
II.
10
5
0
0
0.2
0.4
0.6
Radius (solar radius)
0.8
1
FIG. 2. The solar plasma is constituted mostly by electrons, protons and neutrons. Variation of the number density of electrons
ne (r) (red curve), up-quarks nu (r) (blue curve) and down-quarks
nd (r) (green curve), and the relative variation of up and down quarks
nu (r)/nd (r) (black curve). In the center nu /nd = 1.3 and near the
surface nu /nd = 1.7.
sound speed difference of helioseismic data and standard solar
model is reduced further [37].
Particularly relevant for our study is the radial profile of the
electron, proton and neutron densities inside the Sun, since
these quantities are fundamental ingredients to test the nonstandard neutrino physics theories.
B.
The solar neutrino sources
The neutrino fluxes produced in the nuclear reactions of the
pp chains and CNO cycle have been computed for an updated
version of the solar standard model, as discussed in the previous section. Figure 1 shows the location of the different
neutrino emission regions of the nuclear reactions for an upto-date SSM. In the Sun’s core, the neutrino emission regions
occur in a sequence of shells, following closely the location
of nuclear reactions, orderly arranged in a sequence dependent on their temperature. The helioseismology data and solar
neutrino fluxes guaranties that such neutrino shells are known
with a great accuracy. The leading source of the energy in the
present Sun are the pp chains nuclear reactions, since the CNO
cycle nuclear reactions contribute with less than 2%. The first
reaction of the pp chains is the P P -ν reaction which has the
largest neutrino emission shell, a region that extends from the
centre to 0.30 R . The P eP -ν reaction has a neutrino emission shell which is similar to the P P reaction, but with a shell
of 0.25 R . These nuclear reactions are strongly dependent
on the total luminosity of the star. Alternatively, the neutrino
emission shells of 8 B-ν and 7 Be-ν extend up to 0.15 R and
0.22R . It is interesting to notice that the maximum emission
of neutrinos for the pp chains nuclear reactions, follows an ordered sequence (see figure 1): 8 B-ν, 7 Be-ν, P eP -ν and P P -
4
ν with the maximum emission located at 0.05, 0.06, 0.08 and
0.10 R . The known neutrino emission shells of the different
CNO cycle nuclear reactions are the following ones: 15 O-ν,
17
F -ν and 13 N -ν. These shells are similar to the 8 B-ν emission shell. The 13 N -ν have two independent shells: one in
the Sun’s deepest layers of the core and a second shell located
between 0.12 and 0.25 of R . The emission of neutrinos for
15
O-ν, 17 F -ν and 13 N -ν shells is maximal at 0.04 - 0.05 of
R . The 13 N -ν neutrinos have a second emission maximum
which is located at 0.16 R .
C.
Neutrinos, Electrons and Quarks
The electron density ne (r) = No ρ(r)/µe (r) where µe is
the mean molecular weight per electron, ρ(r) the density of
matter and No the Avogadro’s number. In this model we will
consider the impact on the quarks up and down. Accordingly,
the density of up and down quarks will be computed from a
relation analogous to ne (r), ni (r) = No ρ(r)/µi (r) (with
i = u, d) where µi (r) is the mean molecular weight per quark
given by
−1
3
3
(1)
µi (r) = (1 + δiu )X(r) + Y (r) + Z(r)
2
2
where i = u, d with X + Y + Z = 1. The distribution of electrons and up and down quarks, as a function of the radius of
the Sun for the standard solar model, is shown in figure 2. The
mean molecular weight per quark is dominated by Hydrogen
and Helium since the only other elements included in Z like
Carbon, Nitrogen, Oxygen and heavier elements contribute
with a very small fraction to the solar plasma. Although the
Z-variation can affect the evolution of the star in the way it
affects the radiative transport [37], its impact on the si and
nsi MSW interactions for µi (r), i = u, d) is small since the
relative radial variation between up- and down-quarks due to
Z-variation is not significant.
III.
MODEL OF NEUTRINO PHYSICS OSCILLATIONS
A.
basic neutrino physics
In the Standard Model, neutrinos interact with other particles only via weak standard interactions (si), which are
described by the Lagragian Lsi which can be decomposed
into components describing the charged and neutral interactions [38–40]. Nevertheless, in the current study, we choose
to write the Lagragian Lsi as an effective interaction Lagrangian [40, 41], which at low and intermediate neutrino energies reads
√
Lsi = −2 2GF gpf (ν̄α γρ Lνα ) f¯γ ρ P f
(2)
where f denotes a lepton or a quark, such as the u−quark
and the d−quark, να are the three light neutrinos (with the
subscript α = e, µ, τ ), P is the chiral projector (is equal
to R or L such that R, L ≡ (1 ± γ 5 )/2) and gpf denotes
the strength of the interaction (ns) as defined in the standard model between neutrinos of flavours α and β and the
P -handed component of the fermion f . Specifically, the gpf
f
f
coupling (left- and right- handed coupling, i.e., gL
and gR
)
for the u-quark (and c- and t-quark) to the Z−boson correu
u
sponds to gL
= 1/2 − 2/3 sin2 θw and gR
= −2/3 sin2 θw .
f
Similarly, the gp for the d-quark (and s- and b-quark) corred
d
sponds to gL
= −1/2 + 1/3 sin2 θw and gR
= 1/3 sin2 θw ,
f
d
the gp for the electron corresponds to gL = −1/2 + sin2 θw
d
and gR
= sin2 θw and the gpf for the neutrino (να with
d
d
α = e, µ, τ ) corresponds to gL
= 1/2 and gR
= 0. θw is
the Weinberg mixing angle [42, 43], with a typical value of
sin2 θw ≈ 0.23 [44].
The evolution of a generic neutrino state να ≡ (νe νµ ντ )T
is described by a Schrödinger-like equation [38], that expresses the evolution of the neutrino between the flavour
states [38] with the distance r, from neutrinos that are produced in the Sun’s core until their arrival to the Earth’s neutrino detectors. The equation reads
dνα
= Hνα = (Hv + Hm ) να ,
(3)
dr
where H is the total Hamiltonian, Hv and Hm are the Hamiltonian components expressions for vacuum and in matter
flavour variations, such that Hv ≡ Mν† Mν /2pν where Mν is
the mass matrix of neutrinos (the term proportional to the neutrino momentum pν is omitted here), and Hm is the Hamiltonian (a diagonal matrix of effective potentials) which depends
on the properties of the solar plasma, i.e., the density and composition of the matter, such that Hm = diag(Ve , Vµ , Vτ ).
The flavour evolution is described in terms of the instantaneous eigenstates of the Hamiltonian in matter νm ≡
(ν1m , ν2m , ν3m )T . These eigenstates are related to the flavour
states by the mixing matrix in matter, Um : να = U m νm .
i
B.
The effective matter potential
As neutrinos propagate in the Sun’s interior, they will oscillate between the three flavour states νe , νµ and ντ due to vacuum oscillations, however in the highly dense medium which
is the Sun’s interior, contrary to their propagation in vacuum,
the scattering of neutrinos with other elementary particles,
like electrons, will enhance their oscillation between flavour
states. Indeed, neutrinos propagating in a dense medium like
the Sun (or Earth) have their flavour between states affected
by the coherent forward scattering, i.e., coherent interactions
of the neutrinos with the medium background [38]. The interaction of neutrinos with the medium proceeds through coherent forward elastic Charged-Current (cc) and Neutral-Current
(nc) scatterings, which as usual are represented by the effective potentials Vαcc and Vαnc for each of the three type of neutrinos.
Therefore, at low energies, the potentials can be evaluated
by taking the average of the effective four-fermion Hamiltonian due to exchange of W and Z bosons over the state describing the background medium. Accordingly, for a non-
5
relativistic un-polarized medium, for the effective potential of
νe , νµ and ντ neutrinos, one obtains
Vα = Vαcc + Vαnc ,
(4)
where α = e, µ, τ .
Let us consider that the solar internal medium is mainly
composed of electrons, up-quarks and down-quarks as in protons and neutrons with the corresponding ne (r), nu (r) and
nd (r) local number densities. The contribution to Hm due
to the cc scattering of electron neutrinos νe (produced in the
Sun’s core) propagating in a homogeneous and isotropic gas
of unpolarized electrons (like the electron plasma found in the
Sun’s interior) is given by
√
Vecc = 2GF ne (r)
(5)
where Gf is the Fermi constant. For νµ and ντ , the potential
due to its cc interactions is zero for most of the solar interior
since neither µ’s nor τ ’s are present, therefore,
Vµcc = Vτcc = 0
Generically, for any active neutrino, the Vαcc reads
√
Vαcc = δαe 2GF ne (r)
(6)
(7)
Analogously, one determines Vαnc for any neutrino due to
nc interactions. Since nc interactions are flavour independent, these contributions are the same for neutrinos of all three
flavours. The neutral-current (nc) potential reads
X√
Vαnc =
2GF gvf nf (r)
(8)
f
The effective potentials Vα are due to the coherent interactions of active flavour neutrinos with the medium through
coherent forward elastic weak cc and nc scatterings.
Inside the Sun, as local matter is composed of neutrons,
protons, and electrons, the effective potential Vα for the different neutrino species (including νe neutrinos) has a quite distinct form which depends on the local number densities ne (r),
nu (r) and nd (r), quantities which depend on the chemical
composition (its metalicity Z) of the Sun’s interior. Nevertheless, at first approximation, since electrical neutrality implies locally an equal number density of protons (uud) and
electrons, Vα takes a more simple form (equation 10), as
the nc potential contribution of protons and electrons cancel
each other. Therefore, only neutrons (udd) contribute to Vαnc .
Hence the last two terms of equation (9) can be expressed
as gvn nn (r) to only take into account the quark contribution
for neutrons. In this expression nn (r) is the local density of
neutrons and the gvn is the neutron coupling constant, it follows that g√vn = gvu + 2gvd = −1/2, and equation (9) reads
Vαnc = − 2/2GF nn (r). Now Vα (equation 10) inside the
Sun yields
√
1
Vα = 2GF δαe ne (r) − nn (r) .
(11)
2
where α = e, µ, τ .
As we will discuss later, only effective potential differences
affect the propagation of neutrinos in matter [46], accordingly, one defines the potential difference between two neutrino flavours α and β as
Vαβ = Vα − Vβ ,
(12)
where α, β = e, µ, τ .
where α = e, µ, τ and f = e, u, d. nf (r) is number density of fermions, electrons, up-quarks (u) and down-quarks
(d) as in protons (uud) and neutrons (udd). The factors gvf
are the axial coupling to fermions (gve = −1/2 + 2 sin2 θw ,
gvu = 1/2 − 4/3 sin2 θw and gvd = −1/2 + 2/3 sin2 θw , see
for example Giunti and Chung [45]). Therefore the effective
potential [38] for any active neutrino due to the neutral-current
Vαnc reads
√
Vαnc = 2GF gve ne (r) + gvu nu (r) + gvd nd (r) . (9)
The Sun’s interior is a normal medium composed of nuclei
(protons and neutrons) and electrons. Since the effective potential for muon and tau neutrinos, Vα (with α = µ, τ or a
combination thereof) is due to the neutral current scattering
only (see equation 11), this leads to Vµτ = Vµ − Vτ = 0.
However, as the effective potential for electron neutrinos depends on the neutral and charged current scatterings, in this
case
√
Veα = 2GF ne (r),
(13)
where nu (r) and nd (r) are the analogue of ne (r), i.e., the
number density of up-quarks and down-quarks in the Sun’s
interior.
Using equations (7) and (8) in equation (4), the effective potential for any active neutrino crossing the solar plasma reads
√
Vα = 2GF δαe ne + gve ne + gvu nu + gvd nd . (10)
where α = µ, τ or a combination thereof. Although for the
Sun and Earth only charged current interactions with electrons
are the only effective potential that contributes to the propagation of electron neutrinos, there are other types of non-typical
matter, like the one found in the core of supernovae and in
the early Universe for which the effective potential difference
Vαβ has a much stronger dependence on the properties of the
background plasma [43, 46].
When neutrinos propagate through matter, the forward scattering of neutrinos off the background matter will induce an
index of refraction for neutrinos. This is the exact analogous
to the index of refraction of light travelling through matter.
However, the neutrino index of refraction will depend on the
neutrino flavour, as the background matter contains different
amounts of scatters for the different neutrinos flavours.
C.
Neutrino oscillation data parameters
As shown in the previous section, the neutrino flavour oscillations model is described with the help of 6 mixing parameters all of which are determined from experimental data [47].
6
The quantities are the following ones: the difference of the
squared neutrino masses ∆m221 , ∆m231 , the mixing angles
sin2 θ12 , sin2 θ13 , sin2 θ23 and the CP-violation phase δCP .
The mass square differences and mixing angles are known
with a good accuracy [48, 49]: ∆m231 is obtained from the
experiments of atmospheric neutrinos and ∆m212 is obtained
from solar neutrino experiments.
The mixing angles are not uniformly well defined: θ12 is
obtained from solar neutrino experiments with an excellent
precision; θ23 is obtained from the atmospheric neutrino experiments, this is the mixing angle of the highest value; θ13
has been firstly estimated from the Chooz reactor [50], its
value is very small and was still very uncertain [51]. Nowadays with Daya Bay and Reno, the situation has largely improved [52]. However, present experiments cannot fix the
value of the CP-violation phase [53].
An overall fit to the data obtained from the different neutrino experiments: solar neutrino detectors, accelerators, atmospheric neutrino detectors and nuclear reactor experiments
suggests that the parameters of neutrino oscillations are the
following ones [48, 49]: ∆m231 ∼ 2.457 ± 0.045 10−3 eV 2 or
(∆m231 ∼ −2.449 ± 0.048 10−3 eV 2 ) , ∆m221 ∼ 7.500 ±
0.019 10−5 eV 2 , sin2 θ12 = 0.304 ± 0.013 ,sin2 θ13 =
0.0218±0.001, sin2 θ23 = 0.562±0.032 and δCP = 2π/25 n
with n = 1, · · · , 25.
In the limiting case where the value of the mass differences, ∆m212 or ∆m231 is large, or one of the angles of mixing
(θ12 , θ23 , θ31 ) is small, the theory of three neutrino flavour oscillations reverts to an effective theory of two neutrino flavour
oscillations [53]. Balantekin and Yuksel have shown that the
survival probability of solar neutrinos calculated in a model
with two neutrino flavour oscillations or three neutrino flavour
oscillations have very close values [54].
D.
The survival of electron neutrinos
Mostly motivated by solar neutrino data, the focus of this
work is the study of the propagation of electron neutrinos,
in particular to determine the survival probability of electron
neutrinos Pe (≡ P (νe → νe ) ) arriving on Earth which have
their flavour changed due to vacuum and solar matter oscillations. Luckily, in the Sun Pe takes a particularly simple form,
since the evolution of neutrinos in matter is adiabatic and for
that reason their contribution for Pe can be cast in a similar
manner to the vacuum-oscillation expression. Accordingly,
the standard parametrization of the neutrino mixing matrix
leads to the following survival probability for electron neutrinos, Pe reads
ad
2 m2
Pe = c213 cm2
13 P2 + s13 s13
(14)
where cij = cos θij , sij = sin θij , and P2ad reads
P2ad =
1
m
(1 + cos (2θ12 ) cos (2θ12
)) .
2
(15)
m
m
The matter angles, θ12
and θ13
which depends equally of
the fundamental parameters of neutrino flavour oscillation and
the properties of solar plasma are determined as follows:
m
− The mixing angle θ12
is determined by
?
V12
m
cos (2θ12
) = −q
?2 + (A?−1 sin (2θ ))2
V12
12
12
(16)
?
where the effective potential V12
(E, r) reads
?
V12
(E, r) = c213 − A?−1
12 cos (2θ12 )
(17)
where A?12 = A? /∆m212 and A? = 2EVαβ [38]. The
parameter A? (E, r) contains the effect of matter on
the electron neutrino propagation as defined by Vαβ ,
given by equation (13). In the specific
case of elec√
tron neutrinos,
A
(E,
r)
=
2E
2G
?
F ne (r) (with
√
Veα = 2GF ne (r)).
m
− The mixing angle θ13
, accordingly to Goswami and
Smirnov [55], is determined by
m
sin2 (θ13
) ≈ sin2 (θ13 ) [1 + 2 A?13 ]
(18)
with A?13 = 2EVeo /∆m231 , where Veo is the effective
potential at √
the electron neutrino production radius ro ,
i.e., Veo = 2GF ne (ro ). The value of ro is different
for the different neutrino sources of the pp chains and
CNO cycle.
E.
New Neutrino Physics
In the presence of physics beyond the standard model [e.g.,
56], the neutral current interactions that are flavour diagonal
and universal in the standard model can have a more general
form. Hence, new interactions arise between neutrinos and
matter, which conveniently one defines as non-standard interactions (nsi), these new neutrino interactions with fermions
are described by a new effective lagrangian [e.g., 41, 57, 58].
Accordingly, the classical lagrangian (equation 2) is generalized to take into account these new types of interactions previously forbidden. The new lagrangian reads
√
P
Lnsi = −2 2GF fαβ
(ν̄α γρ Lνβ ) f¯γ ρ P f
(19)
P
where fαβ
is the equivalent of gpf for the standard interactions
(equation 2), which corresponds to the parametrization of the
strength of the non-standard interactions between neutrinos of
flavours α and β and the P -handed component of the fermion
f [e.g., 56]. Without loss of generality we consider only neutrino interactions with up- and down-quarks [e.g., 59]. In
P
the latter Lagrangian, fαβ
corresponds to two classes of nonstandard terms: flavour preserving non-standard terms proP
portional to fαα
(known as non-universal interactions), and
P
flavour changing terms proportional to fαα
with α 6= β.
Since the atoms and ions of the solar medium in which neutrinos propagate are non-relativistic, the vector part of the nsi
operator gives the dominant contribution for the interactions
of the neutrinos with the plasma of the Sun’s interior, in which
7
Vαβ = Ve δαe δβe +
√
2GF
X
fαβ nf (r)
(20)
3
rd(r)
2.5
ru(r)
case the effective nsi coupling can be described by the followL
R
ing combination [57]: fαβ = fαβ
+ fαβ
. These new kinds of
neutrino interactions lead to a new effective potential difference to describe the propagation of neutrinos in matter [59].
Accordingly, the effective potential difference Vαβ is written
as a generalization of the Vαβ obtained in the standard case:
equations 11 and 13. Hence, Vαβ reads
2
f
1.5
fαβ
where
is the strength of nsi interaction of neutrinos with
the medium. A more detailed discussion about the relations
P
between fαβ and fαβ
can be found in Gonzalez-Garcia and
fαβ
1
0
Maltoni [57]. Usually,
is considered as a free parameter
to be adjusted to fit the solar observational data.
In this work, we study only the nsi interactions of electron neutrinos (νe ) with the solar plasma. Accordingly, as is
common practice, we chose to take into account only the nsi
coupling of electron neutrinos with the up-quarks and downquarks of the solar plasma. Among others, Friedland et al.
[60] have shown that the coupling of electron neutrinos with
up-quarks is parametrized by a set of two independent parameters (uN , uD ), and similarly the coupling of electron neutrinos with down-quarks is parametrized by another set of two
independent parameters (dN , dD ). Each of these parameters
corresponds to a linear combination of the original parameters fαβ which defines the strength of the non-standard neutrino interactions with fermions as defined in equation 19. In
the appendix A we show the relation of fD and fN with the
parameters fαβ , for which f is either d or u since in our study
we are only concerned about the interaction with down- and
up-quarks of the solar plasma. A detailed account of the relevance of these quantities can be found in Gonzalez-Garcia and
Maltoni [e.g., 57], Maltoni and Smirnov [e.g., 59], Friedland
et al. [e.g., 60].
As in the case of standard neutrino interactions, for these
nsi interactions the oscillations of neutrino flavour are still
adiabatic, so the probability of electron neutrino survival is
given by equation (14). However, in this case the quantity
cos (2θm ) has been redefined to take into account the new effective matter potential [59], accordingly
m
cos (2θ12
) ≈ −q
?
Vnsi
(21)
?2 + (2r f + A?−1 sin (2θ ))2
Vnsi
f N
12
12
?
where Vnsi
reads
f
?
Vnsi
(E, r) = c213 − A?−1
12 cos (2θ12 ) − 2rf D .
(22)
where rf (r) = nf (r)/ne (r). Figure 3 shows the variation of
the ratios ru (r) and rd (r) inside the star. The rd (r) is smaller
than ru (r) because the star’s composition is dominated by free
protons (ionized hydrogen). As such, for each down-quark
there are two up-quarks.
0.2
0.4
0.6
Radius (solar radius)
0.8
1
FIG. 3. Variation with the solar radius of the ratios ru (r) =
nu (r)/ne (r) (red curve) and rd (r) = nd (r)/ne (r) (blue curve),
and relative variation of up and down quarks nu (r)/nd (r) (black
curve).
F.
The electron neutrino probability survival
The neutrino emission reactions of the pp chains and the
CNO cycle are produced at high temperatures in distinct layers in the Sun’s core. Similarly, the neutrino flavour oscillations occur in the same regions. The average survival probability of electron neutrinos in each nuclear reaction region is
given by
Z R
−1
hPe (E)ij = Nj
Pe (E, r)φj (r)4πρ(r)r2 dr (23)
0
where Nj is a normalization constant given by Nj =
R R
φj (r)4πρ(r)r2 dr and φj (r) is the electron neutrino
0
emission function for the j nuclear reaction. j corresponds to
the following electron neutrino nuclear reactions: P P , P eP ,
8
B, 7 Be, 13 N , 15 O and 17 F . φj (r) defines the location where
neutrinos are produced in each nuclear reaction j for which
the production is maximum in the layer of radius rj (Cf. figure 1). The neutrino fluxes produced by the different nuclear
reactions are sensitive to the local values of the temperature,
molecular weight, density and electronic density. In this study,
we consider that all neutrinos produced in the solar nuclear reactions are of electron flavour as predicted by standard nuclear
physics, therefore, the local density of quarks only affects the
hPe (E)ij by modifying the flavour of electron neutrinos by a
new nsi interaction like the generalized MSW mechanism.
The survival probability of electron neutrinos hPe (E)ij
given by equation (23) is computed using equations (14), (21)
and (22). Figures 4 and 5 show hPe (E)ij for the different
solar neutrino sources, either in the standard MSW or a generalized MSW. The different neutrino interaction models are
described by a specific set of parameters: (uN , uD , dN , dD ).
Figures 4 and 5 top panels show hPe (E)ij for the standard
MSW mechanism in which case all the parameters mentioned above are equal to zero. The other panels of Fig-
8
0.12
0.55
0.1
Pe (E)
0.45
0.4
0.35
0.3
0.25
Pe (E) − Pe,ref (E)
0.5
PP
PeP(*)
HeP
7Be(*)
8B
13N
15O
17F
0.2 −1
10
0.08
0.06
0.04
0.02
0
10
1
10
Energy (MeV)
2
0
10
−1
10
0.55
0
10
1
10
Energy (MeV)
2
10
0.12
0.5
0.1
0.4
0.35
0.3
0.25
PP
PeP(*)
HeP
7Be(*)
8B
13N
15O
17F
0.2 −1
10
Pe (E) − Pe,ref (E)
Pe (E)
0.45
0
10
1
10
Energy (MeV)
0.08
0.06
0.04
0.02
2
10
0
0.55
−1
10
0.5
0.4
0.35
0.3
0.25
0.2 −1
10
1
10
Energy (MeV)
2
10
0.12
PP
PeP(*)
HeP
7Be(*)
8B
13N
15O
17F
0.1
0
10
1
10
Energy (MeV)
2
10
Pe (E) − Pe,ref (E)
Pe (E)
0.45
0
10
0.08
0.06
0.04
0.02
FIG. 4. The survival probability of electron-neutrinos: the Pe curves
correspond to neutrinos produced in the nuclear reactions located at
different solar radius. The three panels correspond to the following
neutrino models of interaction: si−interaction with electrons (top
panel), nsi−interaction with up-quarks with the coupling constants,
uN = −0.30 and uD = −0.22 (middle panel), and nsi−interaction
with down-quarks with coupling constant, dN = −0.16 and dD =
−0.12 (bottom panel). The reference dotted-black curve defines the
survival probability of electron-neutrinos in the centre of the Sun for
which the si− or nsi−MSW flavour oscillation mechanism is maximum. The other coloured curves follow the same colour scheme
shown in Figure 1.
FIG. 5. The survival probability of electron-neutrinos in function
of the neutrino energy for the different regions of emission. The
different panels show the difference between the survival probability of the different electron neutrino sources and the reference curve
(dotted-black curve in Figure 4). The coloured curves follow the
same colour scheme shown in Figures 1 and 4.
ures 4 and 5 correspond to a generalized MSW mechanism
for which the parameters (uN , uD , dN , dD ) can have values
0
−1
10
0
10
1
10
Energy (MeV)
2
10
9
different of zero. Maltoni and Smirnov [59] among others
have shown that only a relatively small ensemble of parameter combinations (uN , uD , dN , dD ) can be accommodated with
the current set of neutrino flux observations. In this study,
for convenience, we choose to focus on neutrino interactions
for which electron neutrinos couples either with up-quarks
(for which dN = dD = 0) or with down-quarks (for which
uN = uD = 0). Specifically, we chose two fiducial sets of values (fN , fD , f = u, d)) of the ensemble of parameters that fits
simultaneously the solar and KamLAND neutrino data sets
with good accuracy. Figure 4 shows the survival probability of electron-neutrinos in the case of the nsi−interaction
for the parameters sets: (uN = −0.30, uD = −0.22) and
(dN = −0.16, dD = −0.12). As discussed by Maltoni and
Smirnov [59] these values correspond to the two parameters that best fit simultaneously the current solar and KamLAND neutrino data sets. Figures 4 and 5 middle panels
show hPe (E)ij for a neutrino up-quark interaction model, and
Figures 4 and 5 bottom panels show hPe (E)ij for a neutrino
down-quark interaction model.
All the different neutrino interaction models have several
common features. In general the hPe (E)ij are very similar for low- and high-energy neutrinos. It is only for neutrinos with intermediate energy that it is possible to distinguish between the different models (Cf. figure 4). For neutrinos in this energy interval it is possible to distinguish two
effects: one relates with the location of the different neutrino
sources, and a second effect relates with the parameter values
(uN , uD , dN , dD ) of the neutrino interaction model. In the former effect, the hPe (E)ij differentiation results from the fact
that φj (r) are located at different solar radius, as shown in
figure 1. Such effect arises equally in si− and nsi− neutrino
interaction models. The second effect occurs only for nsi−
neutrino interaction models, and it is related with the radial
distribution of up- and down-quarks (cf. Figure 2).
This latter effect is shown in figures 4 and 5 for the two
fiducial models adopted in this study: a pure neutrino upquark model (dN = dD = 0) and a pure neutrino down-quark
model (uN = uD = 0). Accordingly, for neutrinos with an
intermediate energy, the hPe (E)ij corresponding to the neutrino up-quark interaction model has an impact of larger amplitude than the neutrino down-quark interaction model (compare middle and bottom panels in figure 5). For instance,
this effect is very significant for the hPe (E)ipp . Nevertheless,
these preliminary results should be interpreted with caution,
since each neutrino source only produces neutrinos within a
limited range of energy, as such the nsi−effect of hPe (E)ij
shown in figure 5 can be significantly reduced in the final neutrino spectrum of some solar neutrino sources. Indeed, we
remind that the neutrinos emitted by φj (r) are limited to a
specific energy range for each j− nuclear reaction. As such
only an energy portion of hPe (E)ij affects the final emitted
neutrino spectrum. This point will be discussed in more detail
in the next section.
IV.
THE SOLAR ELECTRON NEUTRINO SPECTRA
The solar energy spectrum of electron neutrinos from any
specific nuclear reaction is known to be essentially independent of solar parameters, that is, the energy spectrum created
by a specific nuclear reaction is the same independently of
whether neutrinos are produced in an Earth laboratory or in
the core of the Sun. Therefore, the neutrino energy spectrum
of the different nuclear reactions, can be assumed to be equivalent to its Earth laboratory counterpart. A typical example of
such spectra is the 8 B neutrino energy spectrum emitted by the
8
B nuclear reaction of the pp chains in the Sun’s core. This
solar neutrino spectrum has been shown to be equivalent to
several experimental determinations of the 8 B neutrino spectrum [e.g. 61, 62]. Bahcall and Holstein [63], Napolitano et al.
[64], among others, have shown that the 8 B neutrino spectrum
emitted in the Sun’s core is equal to the spectrum measured
in the laboratory, as the surrounding solar plasma does not affect this type of nuclear reaction. The 8 B neutrino spectrum
measured in the laboratory agrees remarkably well with its
theoretical prediction for neutrinos with an energy below 12
MeV, a small difference appearing only for high energy neutrinos. The experimental 8 B neutrino spectrum deduced from
four laboratory experiments shows a difference with the theoretical prediction at most of 1% [62, 65–68]. Accordingly, we
will consider that the electron neutrino energy spectrum of a
solar nuclear reaction at the specific location where these neutrinos are created is identical to the equivalent neutrino spectrum measured in the laboratory.
All neutrinos produced in the Sun’s nuclear reactions are
of electron neutrino type. It is only during the propagation
phase that these neutrinos vary their flavour between electron,
τ and µ. Suitably, we define the original energy spectrum
of electron neutrinos by Ψse (E) and the end energy spectrum
of neutrinos after the neutrino flavour oscillations by Ψ
e (E).
The first spectrum, which is identical to the neutrino spectrum
obtained in an Earth’s laboratory, relates to the neutrinos produced in nuclear reactions (see figure 1). The latter spectrum
corresponds to neutrinos that have their flavour modified by
the vacuum oscillations and the generalized MSW oscillation
mechanism.
Conveniently, the neutrino spectra Ψse (E) and Ψ
e (E) associated to each of the different solar nuclear reactions of the
pp-chain and CNO-cycle are labelled by an unique subscript
j which can take one of these values: P P , HeP , 8 B, 13 N ,
15
O and 17 F . Hence, the two previous neutrino energy spectra have a simple relation, it reads
s
Ψ
e,j (E) = hPe (E)ij Ψe,j (E),
(24)
where hPe (E)ij is the survival probability of an electron neutrino of energy E. Figure 6 shows the shape of several neutrino spectra Ψ
e,j (E). The final neutrino energy spectrum
Ψe,j (E) is significantly different from the original spectrum
Ψse,j (E). Indeed, while Ψse,j (E) depends only on the properties of the nuclear reaction, Ψ
e,j (E) becomes distinct from
Ψse,j (E) due to the contribution of neutrino flavour oscillations. Specifically, the Ψ
e,j (E) depends on the fundamen-
10
(a)νe (P P )
(b)νe (HeP )
(c)νe (8 B)
(d)νe (13 N )
(e)νe (15 O)
(f)νe (17 F )
FIG. 6. pp-chain and cno-cycle energy neutrino spectra: Ψ
e,j (E) is the electron solar neutrino spectrum for the standard MSW effect (red
area); the other colour curves correspond to Ψe,j
(E),
for
which the generalized MSW effect is taken into consideration: nsi−neutrino
interaction with up-quark (blue area), and nsi−neutrino interaction with down-quarks (green area). Ψ
e,j (E) is defined as the probability per
MeV of an electron-neutrino with an energy E. In the calculation of these neutrino spectra we used an up-to-date standard solar model.
tal parameters related with the neutrino vacuum oscillations
through the (generalized) MSW oscillation mechanism, which
depends on the local densities of electrons and quarks, and the
nsi− coupling constants [69]. As discussed previously, all
these effects are taken into account in hPe (E)ij (equation 23).
In this study, we do not include the mono-energetic spectral lines of pp chains nuclear reactions P eP and 7 Be. Although, the previous result (equation 24) also holds for these
two neutrino sources (corresponds to the sources marked with
the subscript (∗ ) in figure 1), we opt for not including them in
this study, since for these neutrino sources other solar plasma
properties contribute to change the shape of the neutrino spectral lines.
Figure 6 shows the spectra of electron neutrinos for some
of the leading nuclear reactions of the Sun’s core. The general
shape of the spectra Ψ
e,j (E) (equation 24) is a combination
of the neutrino spectrum of the nuclear reaction Ψ
e,j (E), and
hPe (E)ij which depends of local density of electrons, downquarks and up-quarks, as well as of the nsi- parameters of
the generalized MSW mechanism. For the specific set of nsiparameters discussed in this study, clearly the HeP and 8 B
neutrino emission shows the larger variation of the shape of
their spectra.
In both cases the interaction of neutrinos with (up- and
down-) quarks leads to neutrino spectra with quite distinct
shapes (blue and green areas in Figure 6) from the ones found
in the standard MSW neutrino interaction (red area in Figure 6). Equally important is the fact that it is possible to distinguish between the two neutrino models of interaction with
quarks, since each model depends differently of the neutrino
energy. The 15 O and 17 F nuclear reactions also show neutrino
spectra with different shapes, although in this case the impact
of the nsi− interactions is much less pronounced than in the
previous case, at least for the current set of parameters. For
the two other nuclear reactions, P P and 13 N , the impact of
the nsi− interactions is very small. This is somehow expected
since the energy of the neutrinos emitted in these nuclear reactions is relatively small. For this neutrino energy range the
flavour oscillations are dominated by vacuum oscillations and
are almost independent of matter oscillations.
11
FIG. 7. The Ψ
(E) is the electron solar neutrino spectrum for
e,8 B
the standard MSW effect (red area) with the error bar computed for
the forthcoming LENA experiment. The error (black lines) in the
spectrum shape is computed assuming the error in the survival probability is Pe (E) ± 0.025, which corresponds to 5 years of the LENA
measurements. The coloured curves follow the same colour scheme
shown in Figure 6. For clarity we have not included the error bar in
the two other curves. Nevertheless, we note that the error bars for the
other curves are identical to this one.
V.
CONCLUSION
In this study we have computed the expected alteration in
the shape of some leading solar neutrino spectra resulting
from neutrinos having a new type of interactions with upand down-quarks, identical to the MSW oscillations of neutrinos with electrons. This new type of matter interaction, also
known as generalized MSW oscillations, depends of the specific properties of the neutrino interaction model but also of
the local thermodynamic properties of the Sun’s interior.
The study shows that the neutrino spectra of the different
solar nuclear reactions have quite distinct sensitivities to the
new neutrino physics. The HeP and 8 B neutrino spectra
have their shapes more affected by the new interaction between neutrinos and quarks. The 15 O and 17 F neutrino spectra also have a small alteration to their shapes, but these effects
are much less pronounced than in the previous case. The impact of new physics in the P P and 13 N spectra is also very
small.
The new generation of neutrinos experiments such as Low
Energy Neutrino Astronomy [LENA, 6], Jiangmen Underground Neutrino Observatory [JUNO, 7] Jinping Neutrino Experiment [Jinping, 10], Deep Underground Neutrino Experiment [DUNE, 8], and NOνA Neutrino Experiment [NOνA,
9], will allow to test some of new neutrino physics theories.
The most promising evidence to discover nsi− in solar neutrino data is the precise measurement of the 8 B spectrum.
Conveniently, we have estimated how the experimental error
of the next generation of detectors like LENA [70] could affect our conclusion. In Figure 7 is show an error bar estima-
tion on the 8 B spectrum computed assuming the error in the
survival probability is Pe (E) ± 0.025, which is the precision
possible to be obtained for the electron neutrino survival probability after 5 years of LENA measurements [70]. Even in a
relatively short period of 5 years of neutrino observations, it is
already possible to find if neutrinos are experiencing flavour
oscillations due to their interaction with quarks. Indeed, the
identification by a future solar neutrino detector of a strong
distortion in the shape of the solar neutrino spectrum, like the
8
B neutrino spectrum compared to the one predicted by the
standard solar model, will constitute a strong indication for
the existence of interactions between neutrinos and quarks in
the Sun’s core. The location and magnitude of the distortion
of the solar spectrum should give us some indication about the
type of interaction (i.e., up- and down-quarks or both).
In conclusion, we have shown that in the near future neutrino spectroscopic measurements will be used to infer the
new interaction between neutrinos and quarks. This will be
an important and totally independent way of testing new neutrino physics interaction models.
ACKNOWLEDGMENTS
This work was supported by grants from Fundação para
a Ciência e Tecnologia and Fundação Calouste Gulbenkian.
The author thanks the anonymous referee for the comments
and suggestions made to this work.
Appendix A: dimensionless parameters encoding the deviation
from standard interactions
The interactions of neutrinos with matter in the theoretical framework of the non-standard model [e.g., 57] can be
described by the Lagrangian term given by equation 19. In
the following, it is assumed that electron neutrinos couples
only with the up-quarks and down-quarks of the solar plasma
(see section III E for details). For convenience, we adopt the
parametrization of Gonzalez-Garcia and Maltoni [57], Maltoni and Smirnov [59], Friedland et al. [60] in which the coupling of electron neutrinos with either up-quarks or downquarks of the solar plasma is parametrized by a set of two
independent parameters (uN , uD ) or (dN , dD ). Accordingly,
the coefficients fD and fN relate to the original parameters
αβ as
fD = c12 s13 Re eiδCP s23 feµ + c23 feτ
−(1 + s213 )c23 s23 Re fµτ
s2 − s213 c223 f
c2
− 13 fee − fµµ + 23
τ τ − fµµ (A1)
2
2
and
+s13 e−iδCP s223 fµτ
fN = c13 c23 feµ − s23 feτ
− c223 fµτ∗ + c23 s23 fτ τ − fµµ (A2)
.
12
As in this work we are only interested in the interaction of neutrinos with the solar plasma, we will consider at a time the fol-
lowing values of f : e, u and d. A detailed discussion about the
relevance of this parametrization can be found in GonzalezGarcia and Maltoni [e.g., 57], Friedland et al. [e.g., 60].
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